一般社団法人 電子情報通信学会
汎光線時空間 (PoTS) 映像学 時限研究専門委員会

第 2 回 PoTS 映像学シンポジウム

One-shot Underwater Active Stereo Through Refractive Parallel Flat Surfaces
Meng-Yu Jennifer KUO† and Shohei NOBUHARA††† Graduate School of Informatics, Kyoto University Yoshida Honmachi, Sakyo, Kyoto, 606-8501
E-mail: †jennifer@vision.kuee.kyoto-u.ac.jp, ††nob@i.kyoto-u.ac.jp
Abstract This paper presents a novel one-shot underwater active stereo system that realizes a dense 3D shapereconstruction of underwater objects. Real-time active scanning in underwater environment has been a diﬃcult task
due to the encountered challenges associated with refraction, scattering, absorption, etc. In the case with refraction,
image-based correspondence search using epipolar geometry is invalid. The main contribution of this paper is thedevelopment of a practical algorithm to estimate the projector pixel locations from an observed image with a projected
grid pattern in underwater environment. More precisely, the one-to-one correspondence between the projector andcamera pixels was successfully established by employing the co-planarity constraint between 3D rays in water emitted
from the camera and the projector.
Key words Underwater Active Stereo, One-shot Structured Light Systems, Refraction

1.

Introduction

3D analysis of underwater objects can be widely used in various applications. For instances, monitoring the development
process of fertilized eggs can be contributed for bioinformatics;
underwater projection mapping can be employed as an educational interactive system in the aquarium, etc. However, 3D
scanning in underwater environment has been a diﬃcult task
due to the encountered challenges associated with refraction,
scattering, absorption, etc., which indicates that we cannot
simply apply the conventional vision techniques for the geometric modeling.
One of the main techniques for 3D shape reconstruction

Fig. 1

Epipolar geometry without refraction

is done by using the corresponding locations among diﬀerent
views. Compared to the passive stereo techniques, the cor-usually arranged in a specific housing in the underwater envi-

respondence search becomes much easier in active stereo sys-ronment, the refraction phenomenon encountered in this hous-

tem which basically consists of a camera and a projector due

ing needs to be addressed into the process of correspondence

to the uniqueness and distinctiveness of the features in theestimation. More precisely, in the case with refraction, suchprojected patterns. In general, we project multiple regularcorrespondence search process is strongly disturbed because

structured light patterns such as Gray-coded patterns, phaseof the depth-dependent refractive distortions, which indicatesshifting Gray-coded patterns, etc. onto the objects to estimatethat the co-planarity which is given by epipolar geometry is

the one-to-one correspondence between camera and projector.

no longer available (Fig. 2). Thus, it is necessary to look for

In order to realize the active scanning in the dynamic envi-

an eﬃcient way to enable this epipolar assumption.

ronment, one-shot active stereo system has become one of the

The objective of this study is to establish a novel one-shot

main techniques [6]〜[9], which remarkably improves the eﬃ-

underwater active stereo system that realizes the dense 3D

ciency of correspondence search.

shape reconstruction of underwater objects.

In contrast to regular structured light patterns, the correspondence search through one-shot pattern techniques in the2.

Related Works

air environment can be solved eﬃciently by epipolar geome-

Studies related to calibration models for underwater camera

try due to the co-planarity among the 3D point of the object

have been developed for many years [2]〜[6], [10], while none

and its corresponding 2D points on camera/projector imageof them provides a practical strategy for calibration and re-planes (Fig. 1). However, since the camera and projector are—1—

sharing such unknowns in order to resolve the ambiguity by a
per-region manner.
In the study of underwater active one-shot 3D scanning,the depth-dependent refractive distortions have been modeled through a polynomial approximation method [6]. In thismethod, it depth-dependently calibrates the parameters associated with the active scanning system with the projection ofa static wave pattern. However, this approximation method
will inevitably cause some errors on the epipolar lines whichindicates that the accuracy of the reconstruction result willdefinitely be influenced. Besides, the depth-dependent calibration parameters will also cause the inconsistency of theFig. 2

Epipolar geometry with refraction

reconstructed shape due to the sparse set of depth values.
In our study, we only considered the refraction occurred inconstruction procedures in the structured light systems(SLS).

underwater environment, and assumed that the rest of the

Besides, some of the proposed models even become invalid due

complex light phenomenon can all be ignored. We would em-

to the problems that occur during the process of correspon-ploy a virtual camera model to handle the depth-dependent

dence search in SLS.

refractive distortion encountered in the refractive medium ge-

In order to handle the refraction problem through parallelometrically, and deal with the problem of image correspon-flat surfaces, Agrawal et al. have proposed a unified calibra-

dence matching through the employment of refractive epipo-tion technique with multiple layers since the geometry of rays

lar geometry with a one-shot pattern. The accuracy of the 3D

in such system corresponds to an axial camera model [1]. In

capturing of underwater objects would also be evaluated.

their work, they derive forward and backward projection equations for their refractive model. However, it requires solving a

3. Calibration

4th degree equation even if there is only one refractive plane,

3. 1

and requires solving a 12th degree equation in the two refrac-

3. 1. 1

tive planes cases for the forward (3D-to-2D) projection, which

A single underwater camera model can be illustrated in Fig.

may be a time-consuming process. Kawahara et al. proposed

3. Since the housing contains two flat surfaces that are parallel

pixel-wise varifocal camera model(PVCM) with an eﬃcient

to each other, we employed a virtual axial camera model [1],underwater forward (3D-to-2D) projection technique throughwhich is placed at the same location as the real underwatertwo parallel flat surfaces as well as a calibration method forcamera and its optical axis is in the same direction along theunderwater projector-camera system [4]. They modeled a non-normal vector n of the refractive parallel flat surfaces, to sim-

central projection of the underwater cameras where the focal

plify the pinhole model without loss of generality (Fig. 4).

Measurement Model and Its Calibration
Single Underwater Camera

lengths of the projection are varied pixel-by-pixel. They also

Based on the underlying geometry of the rays, the refrac-proposed an underwater active-stereo system consisted of two

tive indices and layer thicknesses can be computed easily. As

cameras and one projector that captures the 3D information

shown in Fig. 4, the light path lca − lcg − lcw is derived fromof dynamic objects in water such as swimming fish based on

the observed points pc

their model. As image-based correspondence searching using

tion points pac (i), pgc (i) on housing.

epipolar geometry is invalid in underwater environment, theytion vectors of lcx toward water from the camera center, whereapplied space carving method for the reconstruction procedurex ∈ {a, g, w} represents each media. Denote µa , µg , µw as theinstead.

refractive indices of each media, and τac , τgc as the thicknesses[AC](i) and passed through the intersecDenote ⃗vcx as the direc-

In terms of one-shot SLS for 3D shape reconstruction of fast

between camera center and the first layer of the housing, andmoving objects, a region-wise method that projects only a sin-

the thickness of the housing itself. The Snell’s law can begle grid pattern consisting of sets of parallel color-coded lines

expressed as√in the air environment has been proposed [7]. Compared to
the regular structured light patterns, this grid pattern cannotµax⃗2va + y⃗v2a = µgptell the absolute corresponding points in the projector image
directly on a pixel basis since the one-shot projection encodes

Let rc =p√√

x⃗2vg + y⃗v2g = µwpp√x⃗2vw + y⃗v2w .
p

p

(1)

x2[AC] + y 2[AC] + z 2[AC] . Then the direction
pc

pc

pc

the pixel position up to two unknown shifts. Instead, the pro-⃗cx and the intersection points pac , pgc can be obtained
vectors V

posed method detects continuous regions in the camera image

as follows,
—2—

3. 2

Calibration of Single Underwater Camera

The calibration of our single underwater camera model canbe done based on the method described in [1], which estimate
the parameters of refraction.
We first attach a chessboard pattern on housing in the water[C]side. The 2D points of the chess corners pc
Fig. 3can be detected ina captured camera image. Here we assume the intrinsic cam-Pinhole camera modelera calibration has been done beforehand and thus we know
the camera ray v(i) for each 3D point of the chess corners
qg (i). Denote Rv as the rotation matrix between the virtual
axial camera model and pinhole camera model, and Tv as thetranslation matrix between the virtual axial camera model andthe chessboard pattern. The transformed 3D point qg + Tv ,[C]

the corresponding transformed camera ray Rv pc , and theFig. 4 Axial camera model

normal vector n are all lied on the same plane. Thus, the coplanarity constraint between the 3D points on the chessboard
and the virtual axial camera rays can be shown as follows,((qg + Tv ) · (0, 0, 1)T × Rv p[C]
c

)

= ⃗0.

(3)

By rewriting the above equation, we then have

xp[C] yqg

T 



xp[C]


c


 −x [C] xq 
g 

pc


 y [C] yqg 

 pc


yp[C]


c


 −y [C] xq 
g 

pc


 zp[C] yqg 


c


z
[C]


pc
c

Fig. 5

Measurement model of underwater projector-camera system

⃗ca =
V

( x [AC] y [AC] z [AC] )
p
p
p
c

rc

c

,

,

rc

c

rc

⃗cw =
V

(

z⃗vcg

µ √
g

µw

,

r23

If i points are observed in the captured camera image, bystacking i such equations we can obtain in matrix form as
Ax = 0, where A is a n × 9 matrix. if i >= 9, we will have in
general a unique solution x defined up to a scale factor. The
(2)

τgc ,

solution to Eq (4) is well known as the eigenvector of AT A
associated with the smallest eigenvalue.

x⃗2vg + y⃗v2g √
c

c

x⃗vca

,

x⃗2va + y⃗v2a
c
c

)
√
y⃗vca
µg √ 2x⃗vg + y⃗v2g √, 1 − (x⃗2vw + y⃗v2w ) .
c
c
c
c
µw
x2 + y 2a⃗vc

3. 1. 2

(4)

c

⃗ca c
V
τa ,
z⃗vca
(
)
√
⃗cg = µa x⃗va , µa y⃗va , 1 − (x2 g + y 2g ) ,
V
⃗
vc
⃗
vc
µg c µg c
pgc = pac + r11 t2 − r21 t1 




r21






r12


 r12 t2 − r22 t1  = ⃗0.




r22






r13


 r t −r t 
23 1 
 13 2−zp[C] xqg

pac =

⃗cg
V


r11

a
⃗
vc

Denote pg (xpg , ypg , zpg ) as the intersection point on second
layer of housing. By using Snell’s law, we can describe point
pg as follows,√√x⃗2va + y⃗v2aUnderwater Projector and Camera√

calibration process. Due to the reversibility of the light, theunderwater projector can be viewed as a reversed underwater
camera and hence we can model the rays emitted from the
projector in the same way as described in previous section.

p

z⃗vpa

x⃗2va + y⃗v2a

As shown in Fig. 5, the camera and projector are arranged
in a specific housing. The planar plane in water is used for thep

x2pg + yp2g =

=

p

z

a
⃗
vp

p

√
τac +

µa
τac +
µg

x⃗2vg + y⃗v2g
p

z⃗vpg

√

p

τgc

√

1−

x⃗2va + y⃗v2a
p

µ2
a
µ2
g

p

τgc ,

(x⃗2va + y⃗v2a )pp(5)

where τac , τgc and µa , µg are the thicknesses and the refractive indices of each media. Assume µa = 1 and set rpg =
—3—

√ 2

xpg + yp2g , r⃗vpa =

√

r⃗
va

x⃗2va + y⃗v2a , and tan θa = z pa . Eq (5)
p

p

⃗
vp

then can be rewritten as



T 

rp2g



 − tan θa2  tan θ2 r2a 
a ⃗
vp 


µ2g



 µ2 τ c 2  g a 22
 = rpg r⃗vpa .
2 τac −r⃗v2pa

(a)

(b)

(c)

(d)

(6)

2

τgc

Given m points in the captured image, we can stack all
equations together to obtain in matrix form as Ax = b, where222x = [µ2g , µ2g τac , τac , τgc ]T . Then the linear least-square solu-

Fig. 6tion is given by
x = (AT A)−1 AT b.

(c) region segmentation, (d) unwrapped image.

(7)

Finally, we refine all of the parameters Rv , Tv , τac , τgc , µg by
a re-projection error minimization.
3. 3 Calibration of Underwater Projector-Camera
SystemThe goal of this part is to estimate the relative pose of the
projector and the camera from images with refractive distortions. The projector can be calibrated by providing the 3D
positions of the projected chess corners in water by the camerathat calibrated before (Fig. 5) as described in [4]. In practice,
the overall process can be described as follows,1

An example of color-coded line pattern decoding: (a) projected pattern, (b)camera captured image with decoded ID,

pattern yields a higher eﬃciency for the image-based correspondence search.
In our work, we employed a color-coded line pattern [7]which provides non-unique correspondences up to two degrees
of freedom, one is for horizontal direction and the other one
is for vertical direction.
More precisely, the lines are first decoded into IDs which
are assigned from 0 to 7 based on the principle of de Bruijn
sequence (Fig. 6(a) and Fig. 6(b)). The interpolation is thenemployed between each line for obtaining the values that are

Estimate the relative pose Rc , Tc between the virtual axial

not on the lines. Because of the inevitable ambiguity occurs

camera and the planar chessboard pattern placed in water.

in the periodically encoded lines pattern, it cannot tell theexact corresponding projector pixel location. Instead we de-2

3

Project the other chessboard pattern on the model planefine the part that has continuous interpolated information as

by projector and estimate its 3D positions using the cali-

the same region (Fig. 6(c)), and then unwrap it in each re-brated camera parameters.

gion separately (Fig. 6(d)). A pair of shifting values is thenEstimate the relative pose Rp , Tp between the underwater

added to each pixel in the same region for the estimation ofprojector and the model plane by using 2D-to-3D corre-

exact corresponding coordinates of projector. In other words,

spondences of the projected chess corners.

each pixel in the same region has the corresponding coordinate of the projector with a pair of common unknown shifts

45Compute the thicknesses τap , τgp of each media based on theshiftx , shifty , which are used to adjust the relative position toprinciple of Snell’s law.

the absolute one (Fig. 7).

Derive the relative pose Rp′ , Tp′ between the virtual axial
projector and the model plane.

In the case without refraction, this ambiguity can be solved
easily by utilizing the co-planarity constraint which is given
by epipolar geometry. However, in the case with refraction,

6

Estimate the relative pose Rp2c , Tp2c between the vir-

this co-planarity constraint among the corresponding rays intual projector and the virtual camera by using the pre-the air becomes invalid.

calibrated parameters.
7

Refine all parameters by minimizing the re-projection error.

4. 2

Correspondence Search of Underwater ProjectorCamera System

In order to realize underwater projector-camera correspondence search using the relative corresponding information obtained in the previous section, we employ the co-planarity4.

One-shot Correspondence Search ofUnderwater Projector-Camera Systemconstraint between 3D rays in water. As shown in Fig. 8,
after transforming the two direction vector V⃗cw , V⃗pw emitted

4. 1 One-shot Pattern Decoding

from the camera and the projector, and the direction vector

In contrast to the regular structured light system, one-shotemitted from pgc to pgp in the water to the same coordinate—4—Fig. 7

Fig. 9

Region-wise method using epipolar geometry in the air

Environmental setup

by employing the linear triangulation. Here the 3D points[W ]P (x[W ] P , y[W ] P , z[W ] P )T of the underwater object can be

described asx[W ] P =y[W ] P =w
x⃗vc,pwz⃗vc,pw
y⃗vc,pwz⃗vc,pz[W ] P + xpgc,p ,(9)z[W ] P + ypgc,p .

After rewriting the above equation, we have




x⃗
vw

1

0

c,p
−z w

0

1

−z w

⃗
vc,p
y⃗
w
vc,p

[W ]


(
P (x[W ] P , y[W ] P , z[W ] P )T =

⃗
vc,p

)
xpgc,pypgc,p(10)

Fig. 8Underwater co-planarity

Given m pairs of corresponding points between the camera
and the projector, we then can stack all equations together tosystem (e.g. underwater virtual axial camera model), theseobtain in matrix form as M [W ] P = b, where M is a 4m × 3

three vectors are lied on the same plane. Therefore, the co-matrix. The linear least-square solution is given byplanarity constraint between these three 3D rays in water can[W ]be shown as(

P = (M T M )−1 M T b.

(11)

)

⃗cw · (Rp2c pgp + Tp2c − pgc ) × (Rp2c V⃗pw ) = 0,V

6.

Evaluation

where Rp2c , Tp2c are the transformation matrices from vir-

6. 1

Experimental Setup

tual axial projector to virtual axial camera. The unknown

The environmental setup can be shown in Fig. 9. In our

pgp

system, we used one camera with 1280×960 resolution (Point-on the housing as well as its corresponding direction vectors⃗pw in Eq (8), which are derived from the corresponding point
Vgrey Flea3 FL3-U3-88S2C-C) and one 1080p projector(BenQpp (up + shiftx , vp + shifty ) in the projector image plane. Suchsurface capturing. The projector-camera system was placed

unknown shifts shiftx , shifty can be estimated by solving Eqaround an octagonal water tank of 900 mm diameter. The(8) with a nonlinear optimization method (e.g. Levenberg-

intrinsic calibrations of both camera and projector are done

Marquart algorithm).

beforehand in the air and the relative pose between the cam-

(8)

shifts shiftx , shifty are included in the intersection point

5.

Underwater 3D Shape Reconstruction

MH680) to realize the underwater active stereo system for 3D

era and the projector was calibrated by the method describedin Section 3. The captured object with the projected grid pat-

Up to this point, calibration parameters as well as thetern was placed in the water tank and observed by the camera

image-based correspondences of underwater projector-camera

and the projector via two parallel flat tank surfaces where thesystem required for 3D shape reconstruction have all been es-

thickness is about 30 mm. The light paths would go throughtimated.

air/glass and glass/water.

The procedure of 3D shape reconstruction can be done—5—.

Fig. 10

Captured images with the projections of horizontal Graycoded patterns.

Fig. 12

Captured images with the projections of phase-shifting
Gray-coded patterns

Fig. 11

Captured images with the projections of vertical Graycoded patterns.

6. 2 Underwater 3D shape reconstruction
6. 2. 1

Correspondence Search with Regular Structured
Light Patterns

In this part, we would like to realize 3D shape reconstructionof underwater object by projecting regular structured lightpatterns on the object in order to obtain the ground truth 3D

Fig. 13Ground truth

3D shape reconstruction.
We first projected a single grid pattern onto the surface ofan underwater object and captured it by a camera (Fig. 14).
Such grid pattern is periodically color-coded (8 lines in a cycle)
which indicates that it cannot provide absolute corresponding
projector pixel locations. From the captured image, the continuous regions were detected in both horizontal (y-axis) andshape.
Here we assume the projector has 256 × 256 resolution
and we projected 5-bits Gray-coded pattern and 2-bits phaseshifting Gray- coded patterns on the object. That is, we took
9 images in total (Figs. 10, 11, and 12). Once we obtain the
one-to-one correspondence between camera and projector bydecoding these regular structured light patterns, we then canrealize an underwater 3D shape reconstruction by using themethod described in Section 5 as the ground truth (Fig. 13).

vertical (x-axis) directions as shown in Fig. 15. The relative
correspondences of projector were provided after unwrapping
the values in each region respectively (Fig. 16).
Once all of the relative one-to-one correspondence between
projector and camera were computed, the absolute correspondences of projector could be estimated by employing the coplanarity constraint in water. Finally, the 3D shape reconstruction was done by using the method described in Section
5 (Fig. 17). The computational time of this part took less

6. 2. 2

Correspondence Search with Single Grid Pattern

than 10 seconds on an Intel Core-i7 2.3GHz PC.

In this part, we would like to realize one-shot underwater
—6—

(a)
Fig. 15(b)
Continuous regions detection (a) vertical (x-axis) direction, (b) horizontal (y-axis) direction.

Fig. 14

6. 2. 3

Captured image with static grid pattern

Error Distribution

In our situation, if there are two pixels on the projector image plane that are apart from each other by only one pixel,
then the two corresponding 3D points in water estimated by(a)the method described in Section 5 diﬀer around 1.3 millime-

Fig. 16

ters from each other.

(b)
Relative correspondences of projector (a) vertical (x-axis)
direction, (b) horizontal (y-axis) direction.

In one-shot underwater 3D shape reconstruction, the regionwise error distributions of unknown shifts for both vertical (xaxis) direction and horizontal (y-axis) direction can be shownin Fig. 18, which were obtained by calculating the diﬀerences
between the estimated unknown shifts and the unknown shifts
computed by utilizing the information of ground truth. The
error of unknown shift in projector image ranged from −20
pixels to 32 pixels in vertical (x-axis) direction and ranged
from −10 pixels to 10 pixels in horizontal (y-axis) direction,
which implied that the deviation of the outcome of 3D shapereconstruction would be around 41.6 millimeters at most.

Fig. 17

3D shape reconstruction result using single grid pattern

In order to evaluate the accuracy of such one-shot reconstruction result by the proposed method with a single grid
pattern, we calculated the error distances from the 3D pointsof ground truth in water in millimeters. The 3D points ofthe ground truth were chosen based on the nearest neighborsearch. Fig. 19 illustrates the error distribution of the one-shotreconstruction.

(a)Fig. 18(b)
Region-wise error distribution of unknown shifts (a) vertical (x-axis) direction, (b) horizontal (y-axis) direction.

As shown in Fig. 18 and Fig. 19, the region-wise errors of
unknown shifts caused the error distribution of the reconstruction result region-wise. Though the calibration / line detection errors might be the source of the oﬀset estimation failure,
this region-wise approach is promising due to the fact that it
can be robust to noise by integrating many pixel-wise observations in the same region. Nevertheless, once the noise / error
accumulates too much, the oﬀset estimation might fail for theentire region.

7.

Conclusion

In this paper, we proposed a novel one-shot 3D shape re-Fig. 19

Error distribution of one-shot 3D shape reconstruction.

construction algorithm using a projector-to-camera pair forobjects in water. We mainly dealt with the refraction encoun-by employing the refractive underwater geometry. More pre-tered in underwater environment. We resolved the ambigu-

cisely, the one-to-one correspondence between projector andity that occurs in the one-shot based correspondence searchcamera pixel locations can be established by employing the co-—7—

planarity constraint between 3D rays in water emitted fromthe camera and the projector through an observed image withchine Vision Conference, BMVC 2011, Dundee, UK, August
29 - September 2, 2011. Proceedings, pages 1–11, 2011.

the projected one-shot pattern in underwater environment.
We have employed a virtual camera model to overcome therefractive distortions geometrically and have successfully realized a one-shot 3D shape reconstruction in underwater environment in practice.
However, there are still rooms for improvement and are worthy for future study. For example, the accuracy of the current
result of one-shot underwater 3D shape reconstruction canpossibly be improved by adding some constraints related tothe continuity of the surface of the object during the processof reconstruction. Moreover, the computational time toward
real-time underwater 3D scanning can be improved.
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